| dc.creator | Barnabas, George | |
| dc.creator | Yegnanarayanan, Venkataraman | |
| dc.date | 2023-01-26 | |
| dc.date.accessioned | 2023-03-09T18:10:08Z | |
| dc.date.available | 2023-03-09T18:10:08Z | |
| dc.identifier | https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5689 | |
| dc.identifier | 10.22199/issn.0717-6279-5689 | |
| dc.identifier.uri | https://revistaschilenas.uchile.cl/handle/2250/223048 | |
| dc.description | A graph G(Z, D) with vertex set Z is called an integer distance graph if its edge set is obtained by joining two elements of Z by an edge whenever their absolute difference is a member of D. When D = P or D ⊆ P where P is the set of all prime numbers then we call it a prime distance graph. After establishing the chromatic number of G(Z, P ) as four, Eggleton has classified the collection of graphs as belonging to class i if the chromatic number of G(Z, D) is i. The problem of characterizing the family of graphs belonging to class i when D is of any given size is open for the past few decades. As coloring a prime distance graph is equivalent to producing a prime distance labeling for vertices of G, we have succeeded in giving a prime distance labeling for certain class of all graphs considered here. We have proved that if D = {2, 3, 5, 7, 7th prime, 10th prime, 13th prime, 16th prime, (7 + j)th prime, ..., (4 + j)th prime for any s ∈ N}, then there exists a prime distance graph with distance set D in class 4 and if D = {2, 3, 5, 4th prime, 6th prime, 8th prime, (4 + j)th prime, ..., (2 + j)th prime for any s ∈ N} then there exists a prime distance graphs with distance set D in class 3. Further, we have also obtained some more interesting results that are either general or existential such as a) If D is a specific sequence of integers in arithmetic progression then there exist a prime distance graph with distance set D, b) If G is any prime distance graph in class i for 1 ≤ i ≤ 4 then G × K2 is also a prime distance graph in the respective class i, c) A countable union of disjoint copies of prime distance graph is again a prime distance graph, d) The Middle/Total graph of a path on n vertices is a prime distance graph. In addition we also provide a new different proof for establishing a fact that all cycles are prime distance graph. | en-US |
| dc.format | application/pdf | |
| dc.language | eng | |
| dc.publisher | Universidad Católica del Norte. | en-US |
| dc.relation | https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5689/4241 | |
| dc.rights | Copyright (c) 2023 George Barnabas, Venkataraman Yegnanarayanan | en-US |
| dc.rights | https://creativecommons.org/licenses/by/4.0 | en-US |
| dc.source | Proyecciones (Antofagasta, On line); Vol. 42 No. 1 (2023); 175-204 | en-US |
| dc.source | Proyecciones. Revista de Matemática; Vol. 42 Núm. 1 (2023); 175-204 | es-ES |
| dc.source | 0717-6279 | |
| dc.source | 10.22199/issn.0717-6279-2023-01 | |
| dc.subject | chromatic number | en-US |
| dc.subject | distance graph | en-US |
| dc.subject | prime distance graph | en-US |
| dc.subject | prime | en-US |
| dc.subject | distance labeling | en-US |
| dc.subject | unit distance graph | en-US |
| dc.subject | 05C12 | en-US |
| dc.subject | 05C15 | en-US |
| dc.title | Chromatic coloring of distance graphs, III | en-US |
| dc.type | info:eu-repo/semantics/article | |
| dc.type | info:eu-repo/semantics/publishedVersion | |
| dc.type | Peer-reviewed Article | en-US |
| dc.type | Text | en-US |
